Uniform Regularity Estimates in Parabolic Homogenization

TL;DR

This paper establishes uniform regularity estimates for solutions to parabolic systems with oscillating coefficients, advancing the understanding of homogenization in time-dependent settings.

Contribution

It provides new uniform interior and boundary regularity estimates for parabolic homogenization problems with oscillating coefficients.

Findings

Uniform interior $W^{1,p}$, Hölder, and Lipschitz estimates

Boundary $W^{1,p}$ and Hölder estimates

Uniform $W^{1,p}$ estimates for initial-Dirichlet problems

Abstract

We consider a family of second-order parabolic systems in divergence form with rapidly oscillating and time-dependent coefficients, arising in the theory of homogenization. We obtain uniform interior $W^{1,p}$, H\"older, and Lipschitz estimates as well as boundary $W^{1,p}$ and H\"older estimates, using compactness methods. As a consequence, we establish uniform $W^{1,p}$ estimates for the initial-Dirichlet problems in $C^{1}$ cylinders.